Solve this equation in integers: $x_1^4 + x_2^4 + \cdots + x_{14}^4 = 9999$.

Suppose integers $a$ and $b$ satisfy $ab\equiv -1 \pmod{24}$. Prove $(a + b)$ must be a multiple of $24$.

Compute $9^{50}\pmod{1000}$.

Find the last three digits of $9 + 9^2 + 9^3 + \cdots + 9^{2000}$.

Let $N = 7\times 8\times 9\times 15\times 16\times 17\times 23\times 24\times 25\times 43$. Compute $N\pmod{11}$.

Let $p$ be a prime and $$\frac{a}{b}=\frac{1}{1^2}+\frac{1}{2^2}+\cdots + \frac{1}{(p-1)^2}$$

where $a$ and $b$ are two co-prime positive integers. Show that $p\mid a$.

Show that $\varphi(n)=n/4$ is impossible to hold.

Select nine different digits from $0$ to $9$ to form a two-digit number, a three-digit number and a four-digit number. The sum of these three numbers is $2017$. Which digit is not selected?

Let $p$ be a prime number and $\lfloor{x}\rfloor$ denote the largest integer not exceeding real number $x$. Show that $$C_n^p\equiv\left\lfloor{\frac{n}{p}}\right\rfloor\pmod{p}$$

Show that from any given $m$ integers, it is always possible to select one or more integers such that their sum is a multiple of $m$.

Show that for any positive integer $k$, it always holds that $10^k\equiv 4\pmod{6}$.

Find the remainder when $10^{10}+10^{100}+10^{1000}+\cdots+10^{\overbrace{\scriptsize{10\cdots 0}}^{2018}}$ is divided by $7$.

Solve the following relation in integers: $$x^2 + a^2 = (x+1)^2 + b^2 = (x+2)^2 + c^2 = (x+3)^2 + d^2$$

How many positive integers $N$, less than $2017$, satisfy $$N^{2016^{2016}}\equiv 1\pmod{2017}$$

Let $p$ is an odd prime, compute $1^{p-1}+2^{p-1}+3^{p-1}+\cdots+(p-1)^{p-1}\pmod{p}$.

Let $p$ is an odd prime, compute $1^{p}+2^{p}+3^{p}+\cdots+(p-1)^{p}\pmod{p}$.

Let $m$ and $n$ be two distinct positive integers. Find the minimal value of $(m+n)$ such that the last three digits of $2017^m$ and $2017^n$ are equal.

Find the multiplicative order of $17$ modulo $1000$.

Let $p$ be a prime and $k$ be a positive integer less than $p$. Show that $\binom{p}{k} \equiv 0 \pmod{p}$.

Let $x$ and $y$ be two integers and $p$ be a prime. Show that $$(x+y)^p\equiv x^p + y^p\pmod{p}$$

(Fermat's little theorem) Show that $a^p\equiv a\pmod{p}$ holds if $p$ is a prime.

Find all powers of $2$, such that after deleting its first digit, the new number is also a power of 2. For example, $32$ is such a number because $32=2^5$ and $2=2^1$.

An integer in the form of $F_n=2^{2^n}+1$ where integer $n\ge 1$ is called a Fermat's number. Let $d_n$ be any divisor of $F_n$. Show that $d_n\equiv 1\pmod{2^{n+1}}$.

Assume positive integer $n > 1$ satisfies $n\mid (2^n+1)$, prove $n$ is a multiple of $3$.